Yes, we are pretty awful, and it’s pretty amazing we manage to get through the day. The reason we do is that our brains were not designed to remember long numbers or to calculate square roots, we were designed to …get through the day.

Thus it’s no surprise that we can spot tigers hiding in the shrubbery, and judge someone’s intent from the curl in the corner of their mouth – things computers can’t even dream of!

Amazing Things the Brain Can Do

There are some really remarkable abilities the evolutionary arms race has given us. Consider for a moment how hard it is to teach these skills to a computer:

Theory of mind – our ability to realize that others have motives and intentions and the ability to guess them reasonably well

Inventiveness – our ability to make connections from disparate fields

Much has been said about these skills, and in particular, much value has been placed on theories about our inventiveness – if only we can understand how we invent, we can unleash a torrent of innovation!

The ideas usually run something like this: the human mind is so highly integrated that many concepts are forced to overlay one another so connections are inevitable – while others suggest the mind reviews new learning each night during sleep and tries to spot patterns, suggesting our innovative spark is really just our pattern recognition skill in disguise [1].

While I suspect there is truth to both theories, there is probably more to it than that…

Another Amazing Skill Often Overlooked

Now – if you have ever caught a child being naughty, you may have been lucky enough to see another remarkable human talent…

Lying.

Lying is tricky. Lying requires amazing computation – it needs theory of mind, it requires creativity, and does its invention under pressure.

Lying requires creating an entire alternate reality that fits the evidence but makes you look innocent of all crimes! It’s so hard that young kids don’t always get it quite right, but at some point most of us master the art. Our brains can also be switched to this mode of inventive overdrive in another way: when we attempt to explain incomplete data.

The most common opportunity to fit a narrative to incomplete data is when we recall faded memories – it turns out many of us can bring out our internal Dr. Seuss when recounting our roles in past events.

And because we all like to think of ourselves as pretty darn awesome, our memories cannot contain any information that could contradict this most evident truth. Thus when we recall situations when we did something downright shameful, our brains become positively electrified and we will magic up perfectly good reasons for what we did out of thin air.

Almost everyone can do it. However, if you ask us to write a short bit of utter fiction, our ability instantly vanishes.

Leveraging Brain Power

So the question is this… how can we tap into these remarkable abilities? Do creative people already do it?

To most people, maths is just something we learned in order to avoid being ripped off. To some, maths is an essential tool, helpful in modelling plague outbreaks or cracking encryption ciphers.

However, for an elite few, maths is simply a parallel universe and they are its explorers.

Today let us discuss what I consider perhaps the most beautiful discovery to date. But first, some introductions…

Part 1: Consider, to start, the circle

If you have a wheel a metre across, it will roll out about 3.14159…metres each revolution. This number, which we call π turns out to be some sort of fundamental property of ‘space’.

The Greeks were not very happy about the ‘messiness’ of this number. They preferred numbers that could be expressed as fractions – while 22/7 was close to π, it was not exact and they lost a lot of sleep trying to find a neat way to write π.

Mathematicians have since grudgingly accepted that it cannot be written as a fraction, and indeed it cannot be written down at all because it has no ‘pattern’ and never ends, those digits just keep coming at (almost) random! Here are the first 100…

Understandably, they decided to call this sort of number ‘irrational‘.

Part 2: Consider now, ‘powers’

Mathematicians may work tirelessly on some very pointless looking things, however, they are still fairly lazy when it comes to writing stuff down. They like shorthand. So rather than writing 3+3+3+3+3 they invented ‘multiplication’, giving them 5×3.

Likewise, rather than writing 3x3x3x3x3 they invented ‘powers’, so they could write 35.

Of course they then realized these tricks could be extended past ‘whole’ numbers. 2.5×3 is 7.5. But what about 32.5?

It works of course, the answer turns out to be about 15.59 plus change.

But what does it mean? 32.5 is three, times by itself, 2.5 times! or 3x3x30.5. What on earth is that?

Well it turns out, when you ponder this (maybe I should say, if you ponder this), that 30.5 is the same as √3. So ‘root three’ is three times itself half a time…

[pregnant pause]

Ok, let’s look at it another way

Consider, for example 32x33, which is the same as (3×3)x(3x3x3) which is the same as 3x3x3x3x3 which is the same as 35 , so 3(2+3).

So using that logic…

3 = 31 = 3(0.5+0.5) = 30.5 x 30.5

And what times itself is equal to 3? Well √3! So 30.5 is √3…

It makes sense now, and we can even get used to saying things like 31.9 x 30.1 = 9.

Of course, these fractional powers also commonly yield those ‘messy numbers’, so abhorred by the Greeks. √3 is, roughly:

1.73205080756887729352744634150587236694280525381038062805580…

The logic follows through for negative numbers. 3-2 is just 1/32 which is 1/9.

Part 3. Now consider ‘e’

y=e^x. The slope is always the same as the value! This has the interesting effect that the tangent to the line always intercepts the y axis precisely 1 unit back…

y=e^x

Here is a third sort of messy number, one which the Greeks are probably glad they missed. We have Leonhard Euler to thank for discovering this one.

He noted there was a number ‘e‘ giving an equation of the form y=ex (see the graphs pictured), where the slope of the curve is the same as the height of the curve at each point.

Strange and pointless sounding perhaps but pretty simple. So y=2x doesn’t work, y=3x doesn’t work, but by trial and error you can find a value for a that works, which is, roughly:

Now solve for x. Seems pretty easy, but really you are cheating. There is no number that solves that equation. Really, to solve it you had to ‘invent’ the concept of a negative number.

Ok. Now consider the equation x2 + 1 = 0

Ah. Trickier! However it turns out that we can do the same trick; this time we simply invent another sort of number – the ‘imaginary’ number. Now if you’ve never heard about these numbers before, you may think I’m joking. Alas I am not. This is what mathematicians have been up to for the last few hundred years, just making stuff up as they go along.

So we define i as √-1, or a number, that when multiplied by itself, yields the more respectable -1.

Aside: Just as -1 is a number which, when multiplied by any negative number renders it decent (i.e. positive) once more.

So i2 + 1 = 0 and the equation is solved. It turns out mathematicians were suddenly able to solve loads of really tedious equations using this trick, which made their entire week.

So we have now got i! At last we are ready to put the puzzle together.

Simplicity emerges from the complex…

So, now I ask, what happens if you raise e to the power of i? What does it even mean? It means, e times itself √-1 times. Ouch. Nonsense surely?

Well it works out at roughly 0.540302306 + 0.841470985 i, which is a right mess, something they call, for fairly self-explanatory reasons, a complex number.

So now lets stick our old friend from the circle, π, in there and see what happens:

What is eiπ?

Surely an even bigger mess? I mean its all these messy irrational numbers combined with this home-made imaginary number…